8.5.42 problem 42
Internal
problem
ID
[770]
Book
:
Differential
equations
and
linear
algebra,
3rd
ed.,
Edwards
and
Penney
Section
:
Section
1.6,
Substitution
methods
and
exact
equations.
Page
74
Problem
number
:
42
Date
solved
:
Monday, January 27, 2025 at 03:04:32 AM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _exact, _rational]
\begin{align*} \frac {2 x^{{5}/{2}}-3 y^{{5}/{3}}}{2 x^{{5}/{2}} y^{{2}/{3}}}+\frac {\left (3 y^{{5}/{3}}-2 x^{{5}/{2}}\right ) y^{\prime }}{3 x^{{3}/{2}} y^{{5}/{3}}}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 181
dsolve(1/2*(2*x^(5/2)-3*y(x)^(5/3))/x^(5/2)/y(x)^(2/3)+1/3*(-2*x^(5/2)+3*y(x)^(5/3))*diff(y(x),x)/x^(3/2)/y(x)^(5/3) = 0,y(x), singsol=all)
\begin{align*}
y &= \frac {2^{{3}/{5}} 3^{{2}/{5}} \left (x^{{5}/{2}}\right )^{{3}/{5}}}{3} \\
y &= -\frac {\left (i \sqrt {2}\, \sqrt {5-\sqrt {5}}+\sqrt {5}+1\right )^{3} 2^{{3}/{5}} 3^{{2}/{5}} \left (x^{{5}/{2}}\right )^{{3}/{5}}}{192} \\
y &= \frac {\left (i \sqrt {2}\, \sqrt {5-\sqrt {5}}-\sqrt {5}-1\right )^{3} 2^{{3}/{5}} 3^{{2}/{5}} \left (x^{{5}/{2}}\right )^{{3}/{5}}}{192} \\
y &= -\frac {\left (i \sqrt {2}\, \sqrt {5+\sqrt {5}}-\sqrt {5}+1\right )^{3} 2^{{3}/{5}} 3^{{2}/{5}} \left (x^{{5}/{2}}\right )^{{3}/{5}}}{192} \\
y &= \frac {\left (i \sqrt {2}\, \sqrt {5+\sqrt {5}}+\sqrt {5}-1\right )^{3} 2^{{3}/{5}} 3^{{2}/{5}} \left (x^{{5}/{2}}\right )^{{3}/{5}}}{192} \\
\frac {x}{y^{{2}/{3}}}+\frac {y}{x^{{3}/{2}}}+c_1 &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.075 (sec). Leaf size: 260
DSolve[1/2*(2*x^(5/2)-3*y[x]^(5/3))/x^(5/2)/y[x]^(2/3)+1/3*(-2*x^(5/2)+3*y[x]^(5/3))*D[y[x],x]/x^(3/2)/y[x]^(5/3) == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \left (\frac {2}{3}\right )^{3/5} \left (x^{5/2}\right )^{3/5} \\
y(x)\to c_1 x^{3/2} \\
y(x)\to -\left (-\frac {2}{3}\right )^{3/5} x^{3/2} \\
y(x)\to \left (-\frac {2}{3}\right )^{3/5} x^{3/2} \\
y(x)\to -\left (\frac {2}{3}\right )^{3/5} x^{3/2} \\
y(x)\to \left (\frac {2}{3}\right )^{3/5} x^{3/2} \\
y(x)\to -\sqrt [5]{-1} \left (\frac {2}{3}\right )^{3/5} x^{3/2} \\
y(x)\to \sqrt [5]{-1} \left (\frac {2}{3}\right )^{3/5} x^{3/2} \\
y(x)\to -(-1)^{2/5} \left (\frac {2}{3}\right )^{3/5} x^{3/2} \\
y(x)\to (-1)^{2/5} \left (\frac {2}{3}\right )^{3/5} x^{3/2} \\
y(x)\to -(-1)^{4/5} \left (\frac {2}{3}\right )^{3/5} x^{3/2} \\
y(x)\to (-1)^{4/5} \left (\frac {2}{3}\right )^{3/5} x^{3/2} \\
y(x)\to \left (\frac {2}{3}\right )^{3/5} \left (x^{5/2}\right )^{3/5} \\
\end{align*}